Ways to Remove Edges from a Complete Graph to make Odd Edges
Given a complete graph with N vertices, the task is to count the number of ways to remove edges such that the resulting graph has odd number of edges.
Input: N = 3
The initial graph has 3 edges as it is a complete graph. We can remove edges (1, 2) and (1, 3) or (1, 2) and (2, 3) or (1, 3) and (2, 3) or we do not remove any of the edges.
Input: N = 4
Approach: As the graph is complete so the total number of edges will be E = N * (N – 1) / 2. Now there are two cases,
- If E is even then you have to remove odd number of edges, so the total number of ways will be which is equivalent to .
- If E is odd then you have to remove even number of edges, so the total number of ways will be which is equivalent to .
Note that if N = 1 then the answer will be 0.
Below is the implementation of the above approach:
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